What Data Augmentation Do We Need for Deep-Learning-Based Finance?
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What Data Augmentation Do We Need for Deep-Learning-Based
Finance?
Liu Ziyin1 , Kentaro Minami2 , Kentaro Imajo2
1 Department of Physics, University of Tokyo
2 Preferred Networks, Inc.
June 9, 2021
arXiv:2106.04114v1 [cs.LG] 8 Jun 2021
Abstract
The main task we consider is portfolio construction in a speculative market, a fundamental problem
in modern finance. While various empirical works now exist to explore deep learning in finance, the
theory side is almost non-existent. In this work, we focus on developing a theoretical framework for
understanding the use of data augmentation for deep-learning-based approaches to quantitative finance.
The proposed theory clarifies the role and necessity of data augmentation for√finance; moreover, our
theory motivates a simple algorithm of injecting a random noise of strength ∣rt−1 ∣ to the observed
return rt . This algorithm is shown to work well in practice.
1 Introduction
There is an increasing interest in applying machine learning methods to problems in the finance industry.
This trend has been expected for almost forty years [Fama, 1970], when well-documented and fine-grained
(minute-level) data of stock market prices became available. In fact, the essence of modern finance is fast
and accurate large-scale data analysis [Goodhart and O’Hara, 1997], and it is hard to imagine that machine
learning should not play an increasingly crucial role in this field. In contemporary research, the central
theme in machine-learning based finance is to apply existing deep learning models to financial time-series
prediction problems [Imajo et al., 2020, Ito et al., 2020, Buehler et al., 2019, Jay et al., 2020, Imaki et al.,
2021, Jiang et al., 2017, Fons et al., 2020, Lim et al., 2019, Zhang et al., 2020], which have demonstrated the
hypothesized usefulness of deep learning for the financial industry.
However, one major existing gap in this interdisciplinary field of deep-learning finance is the lack of a
theory relevant to both finance and machine learning. The goal of this work is to propose such a framework,
where machine learning practices are analyzed in a traditional financial-economic utility theory setting. We
then demonstrate the success and the applicability of such a theoretical framework for designing a simple yet
effective data augmentation technique for financial time series. To summarize, our main contributions are
(1) to demonstrate how we can use utility theory to analyze practices of deep-learning-based finance, (2) to
theoretically study the role of data augmentation in the deep-learning-based portfolio construction problem
and (3) to propose a novel and theoretically-motivated machine learning technique for portfolio construction
problems. Organization: the next section discusses the main related works; Section 3 provides the requisite
finance background for understanding this work; Section 4 presents our theoretical contributions, which is
a framework for understanding machine-learning practices in the portfolio construction problem; Section
5 proposes a theoretically motivated algorithm for the task; section 6 validates the proposed theory and
methods with experiments. Also see the beginning of the Appendix for a table of contents.
2 Related Works
Existing deep learning finance methods. In recent years, various empirical approaches to apply state-
of-the-art deep learning methods to finance have been proposed [Imajo et al., 2020, Ito et al., 2020, Buehler
et al., 2019, Jay et al., 2020, Imaki et al., 2021, Jiang et al., 2017, Fons et al., 2020]. The interested readers
are referred to [Ozbayoglu et al., 2020] for detailed descriptions of existing works. However, we notice that
1
one crucial gap is the complete lack of theoretical analysis or motivation in this interdisciplinary field of
AI-finance. This work makes one initial step to bridge this gap. One theme of this work is that finance-
oriented prior knowledge and inductive bias is required to design the relevant algorithms; Ziyin et al. [2020]
shows that incorporating prior knowledge into architecture design is key to the success of neural networks
and applied neural networks with periodic activation functions to the problem of financial index prediction.
Imajo et al. [2020] shows how to incorporate no-transaction prior knowledge into network architecture design
when transaction cost is incorporated.
In fact, most generic and popular machine learning techniques
are proposed and have been tested for standard ML tasks such as
image classification or language processing. Directly applying the
ML methods that work for image tasks is unlikely to work well
for financial tasks, where the nature of the data is different. See
Figure 1, where we show the performance of a neural network di-
rectly trained to maximize wealth return on MSFT during 2019-
2020. Using popular, generic deep learning techniques such as
weight decay or dropout does not result in any improvement over
the baseline. In contrast, the proposed method does. Combin-
ing the proposed method with weight decay has the potential to Figure 1: Performance (measured by the
improve the performance a little further, but the improvement is Sharpe ratio) of various algorithms on MSFT
much lesser than the improvement of using the proposed method (Microsoft) from 2018-2020. Directly apply-
over the baseline. This implies that a generic machine learning ing generic machine learning methods, such
method is unlikely to capture well the inductive biases required as weight decay, fails to improve the vanilla
to tackle a financial task. The present work proposes to fill this model. The proposed method show signifi-
gap by showing how finance knowledge can be incorporated into cant improvement.
algorithm design.
Data augmentation. Consider a training loss function of the additive form L = N1 ∑i `(xi , yi ) for N pairs
of training data points {(xi , yi )}N
i=1 . Data augmentation amounts to defining an underlying data-dependent
distribution and generating new data points stochastically from this underlying distribution. A general way
to define data augmentation is to start with a datum-level training loss and transform it to an expectation
over an augmentation distribution P (z∣(xi , yi )) [Dao et al., 2019], `(xi , yi ) → E(zi ,gi )∼P (z,g∣(xi ,yi )) [`(zi , gi )],
and the total training loss function becomes
1 N
Laug = ∑ E(zi ,gi )∼P (z,g∣(xi ,yi )) [`(zi , gi )]. (1)
N i=1
One common example of data augmentation is injecting isotropic gaussian noise to the input [Shorten and
Khoshgoftaar, 2019, Fons et al., 2020], which is equivalent to setting P (z, g∣(xi , yi )) ∼ δ(g−yi ) exp [−(z − xi )T (z − xi )/(2σ 2 )]
for some specified strength σ 2 . Despite the ubiquity of data augmentation in deep learning, existing works
are often empirical in nature [Fons et al., 2020, Zhong et al., 2020, Shorten and Khoshgoftaar, 2019, Antoniou
et al., 2017]. For a relevant example, Fons et al. [2020] empirically evaluates the effect of different types
of data augmentation in a financial series prediction task. Dao et al. [2019] is one major recent theoretical
work that tries to understand modern data augmentation theoretically; it shows that data augmentation is
approximately learning in a special kernel; He et al. [2019] argues that data augmentation can be seen as an
effective regularization. However, no theoretically motivated data augmentation method for finance exists
yet. One major challenge and achievement of this work is to develop a theory that bridges the traditional
finance theory and machine learning methods. In the next section, we introduce the portfolio theory. A brief
literature review of portfolio theory is given in Section C.
3 Background: Markowitz Portfolio Theory
How to make optimal investment in a financial market is the central concern of portfolio theory. Consider a
market with an equity (a stock) and a fixed-interest rate bond (a government bond). We denote the price
of the equity at time step t as St , and the price return is defined as rt = St+1St−St , which is a random variable
with variance Ct , and the expected return gt ∶= E[rt ]. Our wealth at time step t is Wt = Mt + nt St , where
Mt is the amount of cash we hold, and ni the shares of stock we hold for the i-th stock. As in the standard
2
finance literature, we assume that the shares are infinitely divisible; usually, a positive n denotes holding
(long) and a negative n denotes borrowing (short). The wealth we hold initially is W0 > 0, and we would
like to invest our money on the equity; we denote the relative value of the stock we hold as πt = nW t St
t
; π is
called a portfolio; the central challenge in portfolio theory is to find the best π. At time t, our wealth is Wt ;
after one time step, our wealth changes due to a change in the price of the stock (setting the interest rate
to be 0): ∆Wt ∶= Wt+1 − Wt = Wt πt rt . The goal is to maximize the wealth return Gt ∶= πt ⋅ rt at every time
step while minimizing risk 1 . The risk is defined as the variance of the wealth change:
Rt ∶= R(πt ) ∶= Varrt [Gt ] = (E[rt2 ] − gt2 ) πt2 = πt2 Ct . (2)
The standard way to control risk is to introduce a “risk regularizer” that punishes the portfolios with a large
risk [Markowitz, 1959, Rubinstein, 2002].2 Introducing a parameter λ for the strength of regularization (the
factor of 1/2 appears for convention), we can now write down our objective:
λ
πt∗ = arg max U (π) ∶= arg max [π T Gt − R(π)] . (3)
π π 2
Here, U stands for the utility function; λ can be set to be the desired level of risk-aversion. When gt and Ct is
known, this problem can be explicitly solved. However, one main problem in finance is that its data is highly
limited and we only observe one particular realized data trajectory, and gt and Ct are hard to estimate.
This fact motivates for the necessity of data augmentation and synthetic data generation in finance [Assefa,
2020]. In this paper, we treat the case where there is only one asset to trade in the market, and the task of
utility maximization amounts to finding the best balance between cash-holding and investment. The equity
we are treating is allowed to be a weighted combination of multiple stocks (a portfolio of some public fund
manager, for example), and so our formalism is not limited to single-stock situations. In section E.1, we
discuss portfolio theory with multiple stocks.
4 Portfolio Construction as a Training Objective
Recent advances have shown that the financial objectives can be interpreted as training losses for an appro-
priately inserted neural-network model [Ziyin et al., 2019, Buehler et al., 2019]. It should come as no surprise
that the utility function (3) can be interpreted as a loss function. When the goal is portfolio construction,
we parametrize the portfolio πt = πw (xt ) by a neural network with weights w, and the utility maximization
problem becomes a maximization problem over the weights of the neural network. The time-dependence is
modeled through the input to the network xt , which possibly consists of the available information at time
t for determining the future price3 . The objective function (to be maximized) plus a pre-specified data
augmentation transform xt → zt with underlying distribution p(z∣xt ) is then
1 T
πt∗ = arg max { ∑ Et [Gt (πw (zt ))] − λVart [Gt (πw (zt ))]} , (4)
w T t=1
where Et ∶= Ezt ∼p(z∣xt ) . In this work, we abstract away the details of the neural network to approximate π.
We instead focus on studying the maximizers of this equation, which is a suitable choice when the underlying
model is a neural network because one primary motivation for using neural networks in finance is that they
are universal approximators and are often expected to find such maximizers [Buehler et al., 2019, Imaki
et al., 2021].
The ultimate financial goal is to construct π ∗ such that the utility function is maximized with respect to
the true underlying distribution of St , which can be used as the generalization loss (to be maximized):
πt∗ = arg max {ESt [Gt (π)] − λVarSt [Gt (π)]} . (5)
πt
1 Itis important to not to confuse the price return rt with the wealth return Gt .
2 In principle, any concave function in Gt can be a risk regularizer from classical economic theory [Von Neumann and
Morgenstern, 1947], and our framework can be easily extended to such cases; one common alternative would be R(G) = log(G)
[Kelly Jr, 2011]; our formalism can be extended to such cases.
3 It is helpful to imagine x as, for example, the prices of the stocks in the past 10 days.
t
3
Figure 2: Effect of data augmentation by different noise injections. We see that the proposed data augmentation
scheme preserves the structures in the original financial data, while the baseline methods erase such structures.
First Row: APPLE from 2017 to 2020. Left: Raw Return rt . Mid Left: Noise proportional to rt St2 (the
proposed strength). Mid Right: Noise proportional to 1 (additive noise). Right: Noise proportional to St2 (naive
multiplicative noise). Second Row: Effect on the autocorrelation of the price returns. Left: APPLE daily. Mid
Left: BITCOIN daily. Mid Right: NASDAQ daily. Right: TESLA minutely.
Note the difference in taking the expectation between Eq (4) and (5) is that Et is computed with respect to
the training set we hold, while ESt ∶= ESt ∼p(St ) is computed with respect to the underlying distribution of St
given its previous prices. We used the same short-hands for Vart and VarSt . Technically, the true utility we
defined is an in-sample counterfactual objective, which roughly evaluates the expected utility to be obtained
if we restart from yesterday, which is a relevant measure for financial decision making. In Section 4.5, we
also analyze the out-of-sample performance when the portfolio is static.
4.1 Standard Models of Stock Prices
The expectations in the true objective Equation (5) need to be taken with respect to the true underlying
distribution of the stock price generation process. In general, the price follows the following stochastic process
∆St = f ({Si }ti=1 ) + g({Si }ti=1 )ηt for a zero-mean and unit variance random noise ηt ; the term f reflects the
short-term predictability of the stock price based on past prices, and g reflects the extent of unpredictability
in the price. A key observation in finance is that g is non-stationary (heteroskedastic) and price-dependent
(multiplicative). One model is the geometric Brownian motion (GBM)
St+1 = (1 + r)St + σt St ηt , (6)
which is taken as the minimal standard model of the motion of stock prices [Mandelbrot, 1997, Black
and Scholes, 1973]; this paper also assumes the GBM model as the underlying model. Here, we note
that the theoretical problem we consider can be seen as a discrete-time version of the classical Merton’s
portfolio problem [Merton, 1969]. The more flexible Heston model [Heston, 1993] takes the form dSt =
√
rSt dt + νt St dWt , where νt is the instantaneous volatility that follows its own random walk, and ηt is drawn
from a Gaussian distribution. By definition, GBM is a special case of the Heston model with κ = ξ = 0.
Despite the simplicity of these models, the statistical properties of these models agree well with the known
statistical properties of the real financial markets [Drǎgulescu and Yakovenko, 2002]. The readers are referred
to [Karatzas et al., 1998] for a detailed discussion about the meaning and financial significance of these models.
4.2 No Data Augmentation
In practice, there is no way to observe more than one data point for a given stock at a given time t. This
means that it can be very risky to directly train on the raw observed data since nothing prevents the model
from overfitting to the data. Without additional assumptions, the risk is zero because there is no randomness
in the training set conditioning on the time t. To control this risk, we thus need data augmentation. One
can formalize this intuition through the following proposition, whose proof is given in Section E.3.
4
Proposition 1. (Utility of no-data-augmentation strategy.) Let the price trajectory be generated with GBM
in Eq. (6) with initial price S0 , then the true utility for the no-data-augmentation strategy is
λ 2
Uno−aug = [1 − 2Φ(−r/σ)]r − σ (7)
2
where U (π) is the utility function defined in Eq. (3); Φ is the c.d.f. of a standard normal distribution.
This means that, the larger the volatility σ, the smaller is the utility of the no-data-augmentation strategy.
This is because the model may easily overfit to the data when no data augmentation is used. In the next
section, we discuss the case when a simple data augmentation is used.
4.3 Additive Gaussian Noise to the Training Set
While it is still far from clear how the stock price is correlated with the past prices, it is now well-recognized
that VarSt [St ∣St−1 ] ≠ 0 [Mandelbrot, 1997, Cont, 2001]. This motivates a simple data augmentation technique
to add some randomness to the financial sequence we observe, {S1 , ..., ST +1 }. This section analyzes a vanilla
version of data augmentation of injecting simple Gaussian noise, compared to a more sophisticated data
augmentation method in the next section. Here, we inject random Gaussian noise t ∼ N (0, ρ2 ) to St during
the training process such that zt = St + . Note that the noisified return needs to be carefully defined since
noise might also appear in the denominator, which may cause divergence; to avoid this problem, we define
the noisified return to be r̃t ∶= zt+1St−zt , i.e., we do not add noise to the denominator. Theoretically, we can
find the optimal strength ρ∗ of the gaussian data augmentation to be such that the true utility function is
maximized for a fixed training set. The result can be shown to be
σ 2 ∑t (rt St2 )2
(ρ∗ )2 = . (8)
2r ∑t rt St2
The fact the ρ∗ depends on the prices of the whole trajectory reflects the fact that time-independent data
augmentation is not suitable for a stock price dynamics prescribed by Eq. (6), whose inherent noise σSt ηt is
time-dependent through the dependence on St . Finally, we can plug in the optimal ρ∗ to obtain the optimal
achievable strategy for the additive Gaussian noise augmentation. As before, the above discussion can be
formalized, with the true utility given in the next proposition (proof in Section E.4).
Proposition 2. (Utility of additive Gaussian noise strategy.) Under additive Gaussian noise strategy, and
let other conditions the same as in Proposition 1, the true utility is
r2 (∑t rt St )2
UAdd = ES [ Θ (∑ rt St2 )] , (9)
∑t (rt St )2
t
2λσ 2 T t
where Θ is the Heaviside step function.
4.4 Multiplicative Gaussian Noise Data Augmentation
In this section, we derive a general kind of data augmentation for the price trajectories specified by the GBM
and the Heston model. From the previous discussions, one might expect that a better kind of augmentation
should have ρ = ρ0 St , i.e., the injected noise should be multiplicative; however, we do not start from imposing
ρ → ρSt ; instead, we consider ρ → ρt , i.e., a general time-dependent noise. In the derivation, one can find an
interesting relation for the optimal augmentation strength:
σ2
(ρ∗t+1 )2 + (ρ∗t )2 = rt St2 . (10)
2r
The following proposition gives the true utility of using this data augmentation (derivations in Section E.5).
Proposition 3. (Utility of general multiplicative Gaussian noise strategy.) Under general multiplicative
noise augmentation strategy, and let other conditions the same as in Proposition 1, then the true utility is
r2
Umult = [1 − Φ(−r/σ)]. (11)
2λσ 2
5
Remark. Combining the above propositions, one can quickly obtain that, if σ ≠ 0, then Umult > Uadd and
Umult > Uno−aug with probability 1 (Proof in Section E.6).
Heston Model and Real Price Augmentation. We also consider the more general Heston model. The
derivation proceeds similarly by replacing σ 2 → νt2 ; one arrives at the relation for optimal augmentation:
(ρ∗t+1 )2 + (ρ∗t )2 = 2r
1 2
νt rt St2 . One quantity we do not know is the volatility νt , which has to be estimated
by averaging over the neighboring price returns. One central message from the above results is that one
should add noises with variance proportional to rt St2 to the observed prices for augmenting the training set.
Qualitatively, one can immediately check the effectiveness of this proposed technique. See Figure 2 and its
related discussion in Section 6.1.
4.5 Stationary Portfolio
In the previous sections, we have discussed the case when the portfolio is dynamic (time-dependent). One
slight limitation of the previous theory is that one can only compare the in-sample counterfactual performance
of a dynamic portfolio. Here, we alternatively motivate the proposed data augmentation technique when
the model is a stationary portfolio. One can show that, for a stationary portfolio, the proposed data
augmentation technique gives the overall optimal performance.
Theorem 1. Under the multiplicative data augmentation strategy, the in-sample counterfactual utility and
the out-of-sample utility is optimal among all stationary portfolios.
Remark. See Section E.8 for a detailed discussion and the proof. Stationary portfolios are important in
financial theory and can be shown to be optimal even among all dynamic portfolios in some situations [Cover
and Thomas, 2006, Merton, 1969]. While restricting to stationary portfolios allows us to also compare on
out-of-sample performance, the limitation is that a stationary portfolio is less relevant for a deep learning
model than the dynamical portfolios considered in the previous sections.
4.6 General Theory
So far, we have been analyzing the data augmentation for specific examples of the utility function and the
data augmentation distribution to argue that certain types of data augmentation is preferable. Now we
outline how this formulation can be generalized to deal with a wider range of problems, such as different
utility functions and different data augmentations. For a general utility function U = U (x, π) for some data
point x that describes the current state of the market, and π that describes our strategy in this market state,
we would like to ultimately maximize
max V (π), for V (π) = Ex [U (x, π)] (12)
π
However, only observing finitely many data points, we can only optimize the empirical loss with respect to
some θ−parametrized augmentation distribution Pθ :
1 N
π̂(θ) = arg max ∑ Ezi ∼pθ (z∣xi ) [U (zi , πi )]. (13)
π N i
The problem we would like to solve is to find the effect of using such data augmentation on the true utility
V , and then, if possible, compare different data augmentations and identify the better one. Surprisingly,
this is achievable since V = V (π̂(θ)) is now also dependent on the parameter θ of the data augmentation.
Note that the true utility has to be found with respect to both the sampling over the test points and the
sampling over the N -sized training set:
V (π̂(θ)) = Ex∼p(x) E{xi }N ∼pN (x) [U (x, π̂(θ))] (14)
In principle, this allows one to identify the best data augmentation for the problem at hand:
1 N
θ∗ = arg max V (π̂(θ)) = arg max Ex∼p(x) E{xi }N ∼pN (x) [U (x, arg max ∑ Ezi ∼pθ (z∣xi ) [U (zi , πi )])] , (15)
θ θ π N i
6
and the analysis we performed in the previous sections is simply a special case of obtaining solutions to this
maximization problem. Moreover, one can also compare two different parametric augmentation distributions;
let their parameter be denoted as θα and θβ respectively, then we can say that data augmentation α is better
than β if and only if maxθα V (π̂(θα )) > maxθβ V (π̂(θβ )). This general formulation can also have applicability
outside the field of finance because one can interpret the utility U as a standard machine learning loss function
and π as the model output. This procedure also mimics the procedure of finding a Bayes estimator in the
statistical decision theory [Wasserman, 2013], with θ being the estimator we want to find; we outline an
alternative general formulation to find the “minimax” augmentation in Section E.2.
5 Algorithms
Our results strongly motivate for a specially designed data augmentation for financial data. For a data point
consisting purely of past prices (St , ..., St+L , St+L+1 ) and the associated returns (rt , ..., rt+L−1 , rt+L ), we use
x = (St , ..., St+L ) as the input for our model f , possibly a neural network, and use St+L+1 as the unseen
future price for computing the training loss. Our results suggests that we should randomly noisify both the
input x and St+L+1 at every training step by
⎧ √
⎪
⎪Si → Si + c σ̂i2 ∣ri ∣Si2 i for Si ∈ x;
⎨ √ (16)
⎪
⎪S → St+L+1 + c σ̂i ∣rt+L ∣St+L t+L+1 ;
2 2
⎩ t+L+1
where i are i.i.d. samples from N (0, 1), and c is a hyperparameter to be tuned. While the theory suggests
that c should be 1/2, it is better to make it a tunable-parameter in algorithm design for better flexibility; σ̂t
is the instantaneous volatility, which can be estimated using standard methods in finance [Degiannakis and
Floros, 2015]. One might also assume σ̂ into c.
5.1 Using return as inputs
Practically and theoretically, it is better and standard to use the returns x = (rt , ..., rt+L−1 , rt+L ) as the input,
and the algorithm can be applied in a simpler form:
⎧ √
⎪
⎪ri → ri + c σ̂i2 ∣ri ∣i for ri ∈ x;
⎨ √ (17)
⎪
⎪r → r + c σ̂ 2 ∣r ∣ .
⎩ t+L t+L i t+L t+L+1
5.2 Equivalent Regularization on the output
One additional simplification can be made by noticing the effect of injecting noise to rt+L on the training loss
is equivalent to a regularization. We show in Section D that, under the GBM model, the training objective
can be written as
1 T
arg max { ∑ Ez [Gt (π)] − λc2 σ̂t2 ∣rt ∣πt2 } , (18)
bt T t=1
where the expectation over x is now only taken with respect to the input. This means that the noise
injection on the rt+L is equivalent to adding a L2 regularization on the model output πt . This completes
the main proposed algorithm of this work. We discuss a few potential variants in Section D. Also, it is well
known that the magnitude of ∣rt ∣ has strong time-correlation (i.e., a large ∣rt ∣ suggests a large ∣rt+1 ∣) [Lux
and Marchesi, 2000, Cont and Bouchaud, 1997, Cont, 2007], and this suggests that one can also use the
average of the neighboring returns to smooth the ∣rt ∣ factor in the last term for some time-window of width
τ : ∣rt ∣ → ∣r̂t ∣ = τ1 ∑τ0 ∣rt−τ ∣. In our S&P500 experiments, we use this smoothing technique with τ = 20.
6 Experiments
We demonstrate the strength of the proposed method experimentally. We start with a qualitative comparison
and then move on to benchmark out-of-sample comparisons both on synthetic and real data. We also perform
a classical finance theory based analysis of the proposed method to demonstrate how the proposed method
can be relevant in a finance-theoretic setting.
7
Figure 3: Experiment on geometric brownian motion; S0 = 1, r = 0.005, σ = 0.01. Left: Examples of prices
trajectories in green; the black line shows the expected value of the price. Mid: Comparison on the Sharpe ratio
with baselines. Right: Comparison with other related data augmentation techniques.
6.1 Qualitative Results
See Figure 2, where we compare the augmented data with the raw price return sequence of APPLE from 2017
to 2020. The plotted lines are normalized to have the same overall variance. We see that the theoretically
motivated noise (Mid Left) injection strategy generates an augmented data that is visually closest to the
original, where the well-known structures of volatility clustering [Cont, 2007, Lux and Marchesi, 2000, 1999]
(high-density bumps in the return trajectory) are preserved. In comparison, additive Gaussian noise creates
an imbalance in the data, creating larger randomness early in time (when the price is low). We also compare
with a naive way of making multiplicative noise injection, which only makes the noise proportional to St2
but not rt (related discussion and derivation in Section E.7); we see that the generated data shrinks the
trajectory towards the center and blurs the original structures in the data significantly.
6.2 Benchmark Comparisons
We use the Sharpe ratio as the performance metric (the larger the better). Sharpe ratio is defined as
SRt = √E[∆Wt ] , which is a measure of the profitability per risk. We choose this metric because it is widely
Var[∆Wt ]
accepted as the best single metric for measuring the performance of a financial strategy in theory and practice
[Sharpe, 1966]. Theoretically, it is a classical result in classical financial research that all optimal strategies
must have the same Sharpe ratio [Sharpe, 1964] (also called the efficient capital frontier). Practically, the
reason for comparing on Sharpe ratio is that it is also the most important indicator of success for a fund
manager in the financial industry [Bailey and De Prado, 2014]. For the synthetic tasks, we can generate
arbitrarily many test points to compare the Sharpe ratios unambiguously. We then move to experiments
on real stock price series; the limitation is that the Sharpe ratio needs to be estimated and involves one
additional source of uncertainty.
6.2.1 Geometric Brownian Motion
We first start from experimenting with stock prices generated with a GBM, as specified in Eq. (6), and we
generate a fixed price trajectory with length T = 400 for training; each training point consists of a sequence of
past prices (St , ..., St+9 , St+10 ) where the first ten prices are used as the input to the model, and St+10 is used
for computing the loss. We use a feedforward neural network with the number of neurons 10 → 64 → 64 → 1
with ReLU activations. Training proceeds with the Adam optimizer with a minibatch size of 64.
Results and discussion. See Figure 3. The proposed method is plotted in blue. The middle figure
compares the proposed method with simple training with no data augmentation and training with weight
decay. We see that the proposed method outperforms the baseline methods, suggesting that the proposed
method has incorporated the correct prior knowledge of the problem. The right figure compares the proposed
method with the other two baseline data augmentations we studied in this work. As the theory expects,
the proposed method performs better. We also experiment with the Heston model in the B.5, and similar
results are obtained.
6.2.2 S&P 500 Prices
This section demonstrates the applicability of the proposed algorithm to real market data. In particular,
We use the data from S&P500 from 2016 to 2020, with 1000 days in total. We test on the 365 stocks that
existed on S&P500 from 2000 to 2020. We use the first 800 days as the training set and the last 200 days
8Table 1: Sharpe ratio on S&P 500 by sectors; the larger the better. Best performances in Bold.
Industry Sectors # Stock Merton no aug. weight decay additive aug. naive mult. proposed
Communication Services 9 −0.06±0.04 −0.06±0.04 −0.06±0.27 0.22±0.18 0.20±0.21 0.33±0.16
Consumer Discretionary 39 −0.01±0.03 −0.07±0.03 −0.06±0.10 0.48±0.10 0.41±0.09 0.64±0.08
Consumer Staples 27 0.05±0.03 0.24±0.03 0.23±0.11 0.36±0.08 0.34±0.09 0.35±0.07
Energy 17 0.07±0.03 0.03±0.03 −0.02±0.12 0.70±0.09 0.52±0.10 0.91±0.10
Financials 46 −0.57±0.04 −0.61±0.03 −0.61±0.09 −0.06±0.10 −0.13±0.09 0.18±0.08
Health Care 44 0.23±0.04 0.60±0.04 0.61±0.11 0.86±0.09 0.81±0.09 0.83±0.07
Industrials 44 −0.09±0.03 −0.11±0.03 −0.11±0.08 0.36±0.08 0.28±0.08 0.48±0.08
Information Technology 41 0.41±0.04 0.41±0.04 0.41±0.11 0.67±0.10 0.74±0.11 0.79±0.09
Materials 19 0.07±0.03 0.06±0.03 0.03±0.14 0.47±0.13 0.43±0.13 0.53±0.10
Real Estate 22 −0.14±0.04 −0.39±0.03 −0.40±0.12 0.05±0.10 0.05±0.09 0.19±0.07
Utilities 24 −0.29±0.02 −0.29±0.02 −0.28±0.07 −0.01±0.06 −0.00±0.06 0.15±0.04
S&P500 Avg. 365 −0.02±0.04 −0.00±0.04 −0.01±0.04 0.39±0.03 0.35±0.03 0.51±0.03
for testing. The model and training setting is similar to the previous experiment. We treat each stock as a
single dataset and compare on all of the 365 stocks. Because the full result is too long, We report the Sharpe
ratio per industrial sector (categorized according to GISC) and the average Sharpe ratio of all 365 datasets.
See Section B.1 and B.6 for more detail.
Results and discussion. See Table 1. We see that, without data augmentation, the model works poorly
due to its incapability of assessing the underlying risk. We also notice that weight decay does not improve
the performance (if it is not deteriorating the performance). We hypothesize that this is because weight
decay does not correctly capture the inductive bias that is required to deal with a financial series prediction
task. Using any kind of data augmentation seems to improve upon not using data augmentation. Among
these, the proposed method works the best, possibly due to its better capability of risk control. In this
experiment, we did not allow for short selling; when short selling is allowed, the proposed method also works
the best; see Section B.6. In Section B.6.3, we also perform a case study to demonstrate the capability of
the learned portfolio to avoid a market crash in 2020. We also compare with the Merton’s portfolio [Merton,
1969], which is the classical optimal stationary portfolio constructed from the training data; this method
does not perform well either. This is because the market during the time 2019 − 2020 is volatile and quite
different from the previous years, and a stationary portfolio cannot capture the nuances in the change of the
market condition. This shows that it is also important to leverage the flexibility and generalization property
of the modern neural networks, along side the financial prior knowledge.
6.3 Market Capital Lines
In this section, we link the result we obtained in the previous section
with the concept of market capital line (MCL) in the capital asset
pricing model [Sharpe, 1964], a foundational theory in classical finance.
The MCL of a set of portfolios denotes the line of the best return-risk
combinations when these portfolios are combined with a risk-free asset
such as the government bond; an MCL with smaller slope means better
return and lower risk and is considered to be better than an MCL that
is to the upper left in the return-risk plane. See Figure 4. The risk-free
rate r0 is set to be 0.01, roughly equal to the average 1-year treasury
yield from 2018 to 2020.4 We see that the learned portfolios achieves
a better MCL than the original stocks. The slope of the SP500 MCL
Figure 4: Available portfolios and the
is roughly 0.53, while that of the proposed method is 0.35, i.e., much
market capital line (MCL). The black
better return-risk combinations can be achieved using the proposed dots are the return-risk combinations
method. For example, if we specify the acceptable amount of risk to of the original stocks; the orange dots
be 0.1, then the proposed method can result in roughly 10% more gain are the learned portfolios. The MCL
in annual return than investing in the best stock in the market. This of the proposed method is lower than
example also shows that how tools in classical finance theory can be that of the original stocks, suggesting
used to visualize and better understand the machine learning methods improved return and lower risk.
that are applied to finance, a crucial point that many previous works
lack.
4 https://www.treasury.gov/resource-center/data-chart-center/interest-rates/pages/textview.aspx?data=yield.
97 Outlook
In this work, we have presented a theoretical framework relevant to finance and machine learning to un-
derstand and analyze methods related to deep-learning-based finance. The result is a machine learning
algorithm incorporating prior knowledge about the underlying financial processes. The good performance
of the proposed method agrees with the standard expectation in machine learning that performance can
be improved if the right inductive biases are incorporated. The limitation of the present work is obvious;
we only considered the kinds of data augmentation that takes the form of noise injection. Other kinds of
data augmentation may also be useful to the finance; for example, [Fons et al., 2020] empirically finds that
magnify [Um et al., 2017], time warp [Kamycki et al., 2020], and SPAWNER [Le Guennec et al., 2016] are
helpful for financial series prediction, and there is yet no theoretical understanding of why these methods
suit the financial tasks; a correct theoretical analysis of these methods is likely to advance both the deep-
learning based techniques for finance and our fundamental understanding of the underlying financial and
economic mechanisms. Moreover, A lot is known about finance in the form of “stylized facts” [Cont, 2001],
and these universal empirical facts should also serve as a basis for inspiring more finance-oriented algorithm
design. Meanwhile, our understanding of the underlying financial dynamics is also rapidly advancing; we
foresee better methods to be designed, and it is likely that the proposed method will be replaced by better
algorithms soon. There is potentially positive social effects of this work because it is widely believed that
designing better financial prediction methods can make the economy more efficient by eliminating arbitrage
[Fama, 1970]; the cautionary note is that this work is only for the purpose of academic research, and should
not be taken as an advice for monetary investment, and the readers should evaluate their own risk when
applying the proposed method. Lastly, It is the sincere hope of the authors that this work can attract more
attention to the rapidly growing field of AI-finance.
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12Contents
1 Introduction 1
2 Related Works 1
3 Background: Markowitz Portfolio Theory 2
4 Portfolio Construction as a Training Objective 3
4.1 Standard Models of Stock Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.2 No Data Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.3 Additive Gaussian Noise to the Training Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.4 Multiplicative Gaussian Noise Data Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.5 Stationary Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.6 General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
5 Algorithms 7
5.1 Using return as inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5.2 Equivalent Regularization on the output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
6 Experiments 7
6.1 Qualitative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6.2 Benchmark Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6.2.1 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6.2.2 S&P 500 Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6.3 Market Capital Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
7 Outlook 10
A Experiments 14
B Why not Other Metrics? 14
B.1 Dataset Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
B.2 Sharpe Ratio for S&P500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
B.3 Variance of Sharpe Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
B.4 More on Qualitative Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
B.5 Portfolio Construction Experiments with Heston Model . . . . . . . . . . . . . . . . . . . . . . . 16
B.6 S&P500 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
B.6.1 Underperformance of Weight Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
B.6.2 Additional Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
B.6.3 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
B.6.4 List of Symbols for S&P500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
C Related works 20
D Additional Discussion of the Proposed Algorithm 20
D.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
D.2 Extension to Multi-Asset Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
D.3 Non-Gaussian Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
E Additional Theory and Proofs 22
E.1 Background: Classical Solution to the Portfolio Construction Problem . . . . . . . . . . . . . . 22
E.2 Analogy to Statistical Decision Theory and the Minimax Formulation . . . . . . . . . . . . . . 23
E.3 Proof for no data augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
E.4 Proof for Additive Gaussian noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
E.5 Proof for General Multiplicative Gaussian noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
13E.6 Proof of Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
E.7 Augmentation for a naive multiplicative noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
E.8 Data Augmentation for a Stationary Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
A Experiments
This section describes the additional experiments and the experimental details in the main text. The
experiments are all done on a single TITAN RTX GPU. The S&P500 data is obtained from Alphavantage.5
The code will be released on github.
B Why not Other Metrics?
This paper has used the Sharpe Ratio as the primary metric of objective comparison. A plethora of alternative
metrics in previous works. We discuss a few of the mainly used alternatives and explain why we think they
are unfit for our problem:
Prediction Accuracy of Up-Down Motion of Price. Portfolio construction is not the same as
predicting future price, and it, therefore, does not apply. Even if one cannot forecast future price, portfolio
construction remains a meaningful task for safe investment. Even for the works that do involve direct price
forecasting, we do not recommend using accuracy as the main metric because most of the existing works, to
our knowledge, cannot go beyond 52% in accuracy, and it is quite unconvincing when one tries to compare
an accuracy of 51.5% to an accuracy of 51.7%, for example.
Total Wealth Return: Some existing works compare on total wealth return, but we do not think it
is an appropriate metric for portfolio construction. It is a fundamental result in finance that high return
cannot be achieved simultaneously with low risk, and therefore it is theoretically unjustified and meaningless
to compare total wealth return.
B.1 Dataset Construction
For all the tasks, we observe a single trajectory of a single stock prices S1 , ..., ST . For the toy tasks, T = 400;
−L
for the S&P500 task, T = 800. We then transform this into T − L input-target pairs {(xi , yi )}Ti=1 , where
⎧
⎪
⎪xi = (Si , ..., SL−1 );
⎨ (19)
⎪
⎪y = SL .
⎩ i
xi is used as the input to the model for training; yi is used as the unseen future price for calculating the loss
function. For the toy tasks, L = 10; for the S&P500 task, L = 15. In simple words, we use the most recent L
prices for constructing the next-step portfolio.
B.2 Sharpe Ratio for S&P500
The empirical Sharpe Ratios are calculated in the standard way (for example, it is the same as in [Ito et al.,
2020, Imajo et al., 2020]). Given a trajectory of wealth W1 , ..., WT of a strategy π, the empirical Sharpe
ratio is estimated as
Wi+1
Ri = − 1; (20)
Wi
1 T −1
M̂ = ∑ Ri ; (21)
T i=1
̂ =√ M̂ average wealth return
SR = , (22)
std. of wealth return
∑i=1 Ri2 − M̂ 2
1 T
T
̂ is the reported Sharpe Ratio for S&P500 experiments.
and SR
5 https://www.alphavantage.co/documentation/
14Figure 5: Daily return rt for BITCOIN from 2018 to 2021.
Figure 6: Daily return rt for NASDAQ from 2018 to 2021.
Figure 7: Effect of data augmentation by different noise injections. Left: Raw minutely return rt for TESLA
from 2021 − 4 − 9 to 2021 − 4 − 13. Mid Left: Noise proportional to rt St2 (the proposed strength). Mid
Right: Noise proportional to 1. Right: Noise proportional to St2 .
B.3 Variance of Sharpe Ratio
We do not report an uncertainty for the single stock Sharpe Ratios, but one can easily estimate the uncer-
tainties. The Sharpe Ratio is estimated across a period of T time steps. For the S&P 500 stocks, T = 200,
and by the law of large numbers, the estimated mean M̂ has variance roughly σ 2 /T , where σ is the true
volatility, and so is the estimated standard deviation. Therefore, the estimated Sharpe Ratio can be written
as
̂ =√ M̂
SR (23)
∑i=1 Ri2 − M̂ 2
1 T
T
M + √σT
= (24)
σ + √cT η
M + √σT M 1
≈ = +√ (25)
σ σ T
where and η are zero-mean random√ variables with unit variance. This shows that the uncertainty in the
estimated SR̂ is approximately 1/ T ≈ 0.07 for each of the single stocks, which is often much smaller than
the difference between different methods.
15Figure 8: Experiment on Heston model; S0 = 1, r = 0.005, σ = 0.01, κ = 0.25, θ = 0.04, ρ = 0. Left: Examples
of prices trajectories in green; the black line shows the expected the price at the time. Mid: Comparison
of the proposed data augmentation against other baselines. Right: Comparison against other alternative
noise-based data augmentations. Result obtained after averaging over 500 independent price trajectories.
B.4 More on Qualitative Comparison
Here, we show more visual comparisons of the effects of data augmentation. See Figure 5 for an application
to the daily data on BITCOIN from 2020 to 2021. Again, we see that additive Gaussian noise creates an
imbalance on the trajectory; naive multiplicative noise blurs the structures in the original data, while the
theoretically motivated augmentation is close to the original in structure.
The same conclusion can be reached for the daily return of the NASDAQ index from 2018 to 2021 (last
1000 days) and the minutely data for TESLA. See Figure 6 and Figure 7.
B.5 Portfolio Construction Experiments with Heston Model
Here we compare the case when the underlying price trajectory is generated by a Heston model. As in
[Buehler et al., 2019], we generate the price trajectories with the discretized version of the Heston model.
We generate a fixed price trajectory with length T = 400 for training; each training point consists of a
sequence of past prices (St , ..., St+9 , St+10 ) where the first ten prices are used as the input to the model,
and St+10 is used for computing the loss. We use a feedforward neural network with the number of neurons
10 → 64 → 64 to1 with ReLU as the activation function. Training proceeds with the Adam optimizer with
minibatch size 64. This setting is the same as in Section 6.2.1.
Results and discussion. As can be seen from Figure 8, the result is qualitatively the same as the GBM
case. The proposed method outperforms the baseline methods steadily.
B.6 S&P500 Experiment
This section gives more results and discussion of the S&P 500 experiment.
B.6.1 Underperformance of Weight Decay
This section gives the detail of the comparison made in Figure 1. The experimental setting is the same as
the S&P 500 experiments. For illustration and motivation, we only show the result on MSFT (Microsoft).
Choosing most of the other stocks would give a qualitatively similar plot.
See Figure 1, where we show the performance of directly training a neural network to maximize wealth
return on MSFT during 2018-2020. Using popular, generic deep learning techniques such as weight decay
or dropout does not improve the baseline. In contrast, the proposed method does. Combining the proposed
method with weight decay has the potential to improve the performance a little further, but the improvement
is much lesser than the improvement of using the proposed method over the baseline. This implies that generic
machine learning is unlikely to capture the inductive bias required to process a financial task.
In the plot, we did not interpolate the dropout method between a large p and a small p. The result is
similar to the case of weight decay in our experiments.
16Table 2: Sharpe ratio on S&P 500 when short sell is forbidden. We show the beginning 30 stocks on our list
due to space constraints.
SYMBOL Merton’s Portfolio no augmentation weight decay additive aug. naive multiplicative proposed
A 0.57 0.57 0.57 0.82 0.79 0.80
AAPL 1.55 1.55 1.55 2.56 2.14 1.97
ABC 0.54 0.54 0.54 0.72 0.73 0.65
ABT 0.50 0.50 0.50 0.73 0.56 0.66
ADBE 1.35 1.35 1.35 0.30 1.41 1.47
ADI 0.16 0.16 0.16 0.57 0.48 0.63
ADM −0.41 −0.41 −0.41 0.10 −0.01 0.21
ADP −0.45 −0.45 −0.45 −0.10 −0.18 0.12
ADSK 1.11 1.11 1.11 1.12 1.47 1.53
AEE −0.09 −0.09 −0.09 −0.02 −0.02 0.07
AEP −0.33 −0.33 −0.33 −0.14 −0.21 −0.02
AES −0.17 −0.17 −0.17 0.43 0.40 0.32
AFL −0.80 −0.80 −0.80 −0.14 −0.34 0.14
AIG −1.27 −1.30 −1.30 −0.15 0.14 0.31
AIV −0.70 −0.70 −0.70 −0.18 −0.12 0.17
AJG 0.31 0.31 0.31 0.53 0.51 0.79
AKAM 0.48 0.48 0.48 0.62 0.62 0.45
ALB −0.23 0.59 0.55 0.87 0.77 0.92
ALL −0.02 −0.02 −0.02 0.17 0.28 0.61
ALXN 0.01 0.16 0.05 0.62 0.48 0.53
AMAT 0.57 0.57 0.57 1.01 0.86 1.07
AMD 1.84 1.84 1.84 0.25 2.54 0.51
AME 0.06 0.06 0.06 0.30 0.27 0.65
AMG 0.36 0.37 0.24 0.33 0.41 0.43
AMGN 1.00 1.00 1.00 1.34 1.30 1.19
AMT 0.41 0.41 0.41 0.43 0.59 0.45
AMZN 1.40 1.40 1.40 0.39 1.39 0.97
ANSS 0.97 0.97 0.97 0.55 1.17 1.20
AON 0.30 0.30 0.30 0.75 0.62 0.73
AOS −0.32 −0.32 −0.32 −0.02 −0.13 0.10
APA 0.09 1.32 1.03 1.79 1.60 1.70
APD 0.57 0.57 0.57 0.83 0.87 0.66
APH 0.04 0.04 0.04 0.55 0.39 0.77
ARE 0.05 0.05 0.05 0.28 0.36 0.31
ATVI 1.29 1.29 1.29 0.94 1.20 1.08
S&P500 −0.02 ± 0.04 −0.00 ± 0.04 −0.01 ± 0.04 0.39 ± 0.03 0.35 ± 0.03 0.51 ± 0.03
B.6.2 Additional Results
In this section, we show more results for the S&P500 experiment. See Figure 2. Due to space constraints,
we show the thirty beginning stocks on our list (see the next section). The augmentation strength hyperpa-
rameters for the three kinds of data augmentation is set to be the theoretical optimal values found in the
theory section, given by Eq (8), (97) and (10) respectively. The weight decay hyperparameter is searched
for in the range {10−1 , 10−2 , ..., 10−6 }. For Merton’s portfolio, we do not forbid short selling since it is a
closed-form solution. As shown, using weight decay does not seem to improve the no-data-augmentation
baseline beyond a chance level. We also show the result when short selling is allowed in Figure 3. We see
that the proposed method still performs the best, but the gap and the performance drop. This is possibly
because the market is on average in a growing trend, so forbidding short selling might be a good inductive
bias. In Warren Buffett’s words, “you can’t make big money shorting because the risk of big losses means
you can’t make big bets.”
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